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agroposo wrote:Being a fan of the History and Philosophy of Science, specially astronomy and physics (also mathematics and medicine), I think I have read enough to understand the work of authors like Newton, Kepler, Euler, Lagrange and others.

I didn't know about the work of Dewey Larson and the Reciprocal System theory. So I have spent some time reading his works and I can say that it is difficult to find a sentence understandable, and that there is no mathematical reasoning at all, only gibberish, much like Miles Mathis' works.

I only referenced things where there is an intellectual debt, howsoever slight, but the bulk of the arguments are not in the writeup copied above. They are in the three papers that Hoi attached, and they require no more than high-school algebra and some elementary calculus to follow. If you have read the work of Lagrange, you will have no trouble at all with this.

agroposo wrote:But really, I think all the scientists that propose such new bizarre theories, do not deserve our attention

Yes, but what if Newton's theory is precisely the bizarre theory, that has helped crop up the entirety of modern "space travel"? Does that not deserve attention?

impressive work you have done, I would say. Everyone who has read Newton's Principia knows how difficult is to follow his reasonings and demonstrations, so even harder is to find an error in his works!

But I'm not a scholar and don't have the time or resources to refute your findings concerning Newton's errors. In my limited knowledge (I finished my degree in theoretical physics 30 years ago and I don't work in that field), what I find more disturbing is when you say that an infinite quantity is equal to another infinite quantity. How can that be?

3. The number of rotational forces that act in the direction of the velocity (F’’’, F (7), F (11), F (15)… etc.)are equal to the number of rotational forces acting opposite to that (F’, F (5), F (9), F (13)… etc.)

Last edited by agraposo on May 25th, 2017, 8:30 am, edited 1 time in total.

In the late 19th century, one of the French mathematicians – Henri Poincaré – had already discovered that many ofthe terms being used in the “perturbation” series by mathematicians like Laplace and Lagrange were becominginfinite for long periods of time, making the system unstable. In simple words, the solutions ‘blow up’ fairlyquickly.

Could you please provide any reference to this statement? I thought that Laplace had proved that the Solar System is stable.

It is presently particularly important to understand what are the possible past evolutions of the obliquity (and thus climate) of Mars in the past, as the recent Martian spacecrafts provide very detailed observations of the Martian surface (Malin and Edgett, 2001), and give some accurate account of the weather on the planet.

The first striking example of chaotic behavior in the solar system was given by the chaotic tumbling of Hyperion, a small satellite of Saturn whichstrange rotational behavior was detected during the encounter of the Voyager spacecraft with Saturn (Wisdom, Peale, Mignard, 1984).

SCIENTIFIC AMERICAN DECEMBER, 1986 VOL. 254 NO. 12, 46-57. wrote:The French mathematician Pierre Simon de Laplace proposed that the laws of nature imply strict determinism and complete predictability, although imperfections in observations make the introduction of probabilistic theory necessary. The quotation from Poincaré foreshadows the contemporary view that arbitrarily small uncertainties in the state of a system may be amplified in time and so predictions of the distant future cannot be made... It is the exponential amplification of errors due to chaotic dynamics that provides the second reason for Laplace's undoing.

Chaos theory has been quite well established over the past century since Poincare.

I am unsure what to do with the references you provided about Voyager and Martian spacecraft. Didn't you say "How can anybody follow an astronomer argument seriously when they are referring constantly to the Hubble Space Telescope, or the NASA probes in outer space?" I have the same problem, a lot of these details provided from the spacecraft have to be taken on faith.

Giving those references about spacecrafts, that appear in scientific works, I mean that the data and images purportedly collected from those spacecrafts must be necessarily computer generated or collected from the Earth, or plainly fabricated.

agroposo wrote:what I find more disturbing is when you say that an infinite quantity is equal to another infinite quantity. How can that be?

It is not the comparison of infinite quantities, but infinite series. You can have an infinite series that goes to a finite value.

Even if the series did go to infinity, the comparison is still valid: it is like comparing the number of odd numbers and even numbers:1,3,5,7,9...2,4,6,8,10...

1,3,5,7... and 2,4,6,8... are examples of infinite arithmetic sequences, not series. A series is a sequence that is summed such as 1+3+5+7... and 2+4+6+8... An infinite arithmetic series can only be summed when the d value is between 0 and 1 because numbers get smaller that are to be added and it's limit can be clearly seen when graphing. In the case you have provided both of the sums would be infinite because the d value is 2. When the d value is greater than one the terms increase so when they are added their sums go to infinity. In the examples you have provided, infinity is the sum for both 1+3+5+7... and 2+4+6+8... and so they cannot be equal to each other. Infinity cannot equal infinity because infinity is uncountable so no comparisons can be made.

Now, if Gopi meant sequences instead of series, then there is no sigma value to sum, and therefore no values to compare.

Agroposo, your concerns about how 2 values of infinity or 2 sums of infinite arithmetic series cannot be compared are still valid and Gopi has not sufficiently answered them.

Now, if Gopi meant sequences instead of series, then there is no sigma value to sum, and therefore no values to compare.

That is precisely the meaning, it simply shows that there are an infinite number of terms in the all the sequences.

Note: All it takes is ONE extra term to remove the conceptual basis for the inverse square law. Comparing infinities shows how far off the mark it is from Newton's law, but besides that the actual summation of the series is NOT what I am doing here.

Comparing infinities shows how far off the mark it is from Newton's law

One cannot compare infinities. By definition, infinity is uncountable and therefore not comparable.

Thank you Kham for your logical, mathematical insights. It's good that we have this peer review.

To get beyond the disagreement, maybe we can explain why Gopi is in error. That is: let us frame the principle behind why comparing infinities has been defined with that rule. Why does the rule exist? It's not arbitrary and it comes from sound mathematical principles. (Otherwise, we wouldn't in math discuss definitions like "real numbers" and "unreal numbers" for example).

The logical reason may help us explore if Gopi can communicate differently about his idea without it sounding like his request for us to consider the impossible. My guess is that Gopi accidentally mixed prosaic language with math language, therefore failing to communicate his idea in established understandings/agreements.

I didn't presume that the word series as used by Gopi meant a sum of elements, because nowhere in his paper he is summing the higher order forces, but after re-reading Gopi's paper, I've come to the conclusion that he is using the term series to indicate a set, as when he says in the paper's abstract "an infinite series of higher order rotational forces", meaning that the force F and its derivatives F', F'', F''', F(4), F(5), ..., constitute a set. If we consider all the derivatives, then we have an infinite set.

This infinite set of derivative functions has a one-to-one correspondence with the natural numbers, 1,2,3,4,5, ..., so it is a countable set, because it has the same cardinality as the set of the natural numbers. The cardinality or number of elements of these infinite sets is then a transfinite number, i.e. larger than all the finite numbers, and it is called Aleph zero.

So, when Gopi says that "the number of centripetal forces (F, F(4), F(8), F(12), ... etc.) are equal to the number of centrifugal forces (F'', F(6), F(10), F(14), ... etc.)", this sentence has mathematical meaning only in set theory, because the cardinality of both sets is Aleph zero.

Now, I have another question for Gopi:

If you're postulating that there are "infinite higher order forces", necessary to account for circular and elliptical motion, what would happen if at some point in the sequence, let's say n, the derivative F(n) is zero? Then all the subsequent derivatives would be zero, and your postulate will be contradicted.

In other words, what kind of function for F will permit that the derivatives don't go to zero? I can only think of exponentials or trigonometric functions, as in a Fourier series.

Edit: if all these problems on infinites, series and derivatives seem unsurmountable, then Newton was absolutely right to attack the problem geometrically, like the ancients, although he invented the fluxions. A really clever guy!